Greedy Approximation in Banach Spaces and Compressed Sensing

压缩感知是一种最新的理论，其目标是仅使用少量测量来逼近信号。这是因为许多真实世界的信号可以很好地被稀疏信号逼近或识别。拟议的研究将利用基于冗余系统(词典)的非线性稀疏表示的技术。研究人员的初步结果表明，尽管压缩感知的发展已经在Hilbert空间中发生，但如果我们在Banach空间中考虑这一问题，那么在构造有效的稀疏信号精确恢复算法方面可以有实质性的改进。将有效算法的搜索范围从Hilbert空间扩展到Banach空间是本方案的主要创新之处。我们打算研究属于贪婪算法家族的特定恢复方法的性质。这些方法允许我们以经济的方式构建稀疏表示。这项拟议的研究将开发实用的、可实现的算法，这些算法在收敛、收敛速度和精确恢复方面都被证明是有效的。对于科学家和其他分析师来说，这十年的巨大挑战涉及设计系统来高效地分析数据并从大数据集中提取重要信息。它在国防和民用部门都有无数现有的和潜在的应用。例如，管理大型数据库，如通过监视获得的安全数据库，需要对数据进行分类和压缩，以便于提取重要特征或具体信息。更广泛地说，大数据集的压缩和去噪问题自然而然地出现在生物、医学和许多其他领域。研究数据处理这一方面的科学学科被称为“压缩感觉”。这项拟议的研究的目标是建立一个基本的数学理论，显著提高我们的处理能力(压缩、去噪等)。大数据集。实现这一目标的主要技术是基于贪婪近似方法(贪婪算法)产生的非线性稀疏表示，它允许我们经济地构建稀疏表示。这项提议的目标是利用分析和应用数学中的基本概念，明确定义和量化数据处理挑战，并设计新的、更有效的技术(贪婪算法)来解决这一挑战。我们相信，科学家和其他分析师将能够实施我们的结果，比以前更快、更准确地在大型数据集中找到重要信息。

Compressed sensing is a recent theory that aims to approximate a signal using only a small number of measurements. It is motivated by the fact that many real-world signals can be well-approximated by or identified with sparse signals.The proposed research will utilize techniques based on nonlinear sparse representations with respect to redundant systems (dictionaries). The investigator's preliminary results show that, although the prior development of compressed sensing has occurred in Hilbert spaces, substantial improvement can be made in the construction of efficient algorithms for exact recovery of sparse signals when we consider the problem in a Banach space instead. To widen the search for efficient algorithms from Hilbert spaces to Banach spaces is the main fundamentally new idea of this proposal. We intend to study properties of specific methods of recovery that belong to a family of greedy algorithms. These methods allow us to build sparse representations economically. The proposed research will develop practical, implementable algorithms that are provably efficient with respect to convergence, rate of convergence, and exact recovery.For scientists and other analysts, the great challenges of this decade involve designing systems to efficiently analyze data and extract essential information from large data sets. It has a myriad of existing and potential applications in both the defense and civilian sectors. For example, managing large data bases, such as security data bases obtained through surveillance, requires classification and compression of the data in order to facilitate the extraction of significant features or specific information. More generally, the problem of compression and denoising of large data sets arises naturally in biology, medicine, and many other fields. The scientific discipline that studies this aspect of data processing is called 'compressed sensing.' The goal of the proposed research is to build a fundamental mathematical theory that will significantly increase our ability to process (compress, denoise, etc.) large data sets. The main technique that will be used to achieve this goal is based on nonlinear sparse representations arising from greedy approximation methods (greedy algorithms), which allow us to build sparse representations economically. It is the goal of this proposal to utilize fundamental concepts in analysis and applied mathematics to clearly define and quantify the data processing challenge and design new, more efficient techniques (greedy algorithms) to address it. We believe that scientists and other analysts will be able to implement our results to find essential information in large data sets more quickly and more accurately than before.